Notation

Symbol				Definition
${\mathcal {H}}\,\!$				Hilbert Space
${\mathbb {C} }\,\!$				The set of complex numbers
${\mathbb {R} }\,\!$				The set of real numbers
		Chapter 2 - Qubits and Collections of Qubits
$H\,\!$				The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
$X\,\!$				The Pauli X matrix (see Table 2.1)
$Y\,\!$				The Pauli Y matrix (see Table 2.1)
$Z\,\!$				The Pauli Z matrix (see Table 2.1)
${\mathbb {I} }\,\!$				Identity operator
${\delta _{ij}}\,\!$				The Kronecker delta (see Equation C.17)
${\epsilon _{ijk}}\,\!$				Epsilon (see Equation C.8)
$\otimes \,\!$				The Tensor Product (see Section C.7)
		Chapter 3 - Physics of Quantum Information
$H\,\!$				The Hamiltonian (see Eqs. 3.1 - 3.4)
$\hbar \,\!$				H-bar (Planck's constant divided by $2\pi \,\!$)
$U\,\!$				The Unitary Matrix (see Eqs. 3.5 - 3.6)
$\rho \,\!$				The Density Matrix or Density Operator (see Appendix E - Density Operator: Extensions)
${\mbox{Tr}}\,\!$				The trace of a matrix (see Section C.3.5)
$\det(N)\,\!$				The determinant of a matrix (see Section C.3.6)
${\vec {n}}\,\!$				A unit vector (see Section C.5.1)
$\lambda _{\pm }$				Eigenvalues (see Section C.6)
$\langle {\mathcal {O}}\rangle$				The expectation value of an operator (see Eqs. 3.47 - 3.48)
		Chapter 4 - Entanglement
$\lambda \,\!$				A possible hidden variable (see Eqs. 4.4 - 4.9)
$L\,\!$				Local operations (see Eq. 4.11)
$L\left\vert \Phi \right\rangle$				Local transformations (see Eq. 4.12)
$\left\vert \psi _{+}\right\rangle ,\left\vert \psi _{-}\right\rangle ,$
$\left\vert \phi _{+}\right\rangle ,\left\vert \phi _{-}\right\rangle$				Bell States (see Eq. 4.14)
${\mbox{Tr}}_{B}(\rho _{ss})\,\!$				The partial trace over one of the subsystems (particle states) of a composite system (see Eq. 4.19)
		Chapter 5 - Quantum Information
$\sigma _{i}^{2}\,\!$				The variance of an observable (see Eq. 5.4)
$H\,\!$				The Hadamard gate (see Section 2.3.2, Section 5.4, Eq. 2.16, and Eq. 5.10)
$\left\vert H\right\rangle \,\!$, $\left\vert V\right\rangle \,\!$
$\left\vert D\right\rangle \,\!$, $\left\vert -D\right\rangle \,\!$				Polarization states of photons (see Figure 5.1 and Table 5.1)
		Chapter 6 - Noise in Quantum Systems
$\rho ^{\prime }\,\!$, $\rho \,\!$				Linear mapping vectors (see Eqs. 6.1 - 6.14)
$A\,\!$				The mapping matrix (see Eqs. 6.6 - 6.12)
$H_{S}\,\!$				The Hamiltonian for the system alone (see Eq. 6.15)
$H_{B}\,\!$				The Hamiltonian for the bath alone (see Eq. 6.15)
$S_{\gamma }\,\!$				Operator on the system (see Eq. 6.15)
$B_{\gamma }\,\!$				Operator on the bath (see Eq. 6.15)
$\rho _{S}(0)\,\!$				The initial density matrix of the (open) system (see Eq. 6.16)
$\rho _{B}(0)\,\!$				The initial density matrix of the bath (see Eq. 6.16)
		Chapter 7 - Quantum Error Correcting Codes
$A_{\alpha }\,\!$				Operator element
$P\,\!$				A projector onto the code space
$C(n,t)\;\!$				The binomial coefficient (see Eq. 7.17)
$m\;\!$				The number of code words (see Eq. 7.17)
${\mathcal {S}}\,\!$				The stabilizer
${\mathcal {S}}\subset {\mathcal {P}}_{n}\,\!$				An abelian subgroup of the Pauli group that does not contain $-\mathbb {I} ,\pm i\mathbb {I} \,\!$
${\mathcal {C}}({\mathcal {S}})\,\!$				A stabilizer code
$P_{1}\,\!$				Classical parity check matrix (see Eq. 7.21)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\,\!}				The generator matrix (see Eq. 7.22)
		Chapter 8 - Decoherence-Free/Noiseless Subsystems
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_B\,\!}				Hamiltonian acting only on the bath (see Eq. 8.1)
$H_{I}\,\!$				The interaction Hamilton (see Eqs. 8.1 and 8.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}\,\!}				Denotes the "error algebra" generated by the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{H_S,S_\alpha\}\,\!}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_\alpha\,\!}				Set of error operators acting only on the system (see Eq. 8.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_\alpha\,\!}				Acts only on the bath (see Eq. 8.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{dns}\,\!}				A particular unitary transformation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{cpe}\,\!}				Collective phase error Hamiltonian (see Eq. 8.3)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z^{(i)}\,\!}				A phase operator which acts on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\,\!} th qubit (see Eq. 8.3)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_x\,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_y\,\!} , $S_{z}\,\!$				The three collective errors (see Eq. 8.9)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{dns} \,\!}				Unitary transformation corresponding to the collection of Wigner-Clebsch-Gordan coefficients
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{C}^\perp\,\!}				The orthogonal subspace (to the code) (see Eq. 8.17)
$\Psi \,\!$				An encoded state
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\Lambda_i\}\,\!}				Basis for the noise operators
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\in\mathcal{S} \,\!}				The stabilizer element (see Eq. 8.18)
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_\alpha \,\!}				Elements of the Lie algebra (see Eqs. 8.22 - 8.26, Eq. D.21, Section D.7.1 and Sections D.8.1 - D.8.3)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\,\!}				The Hamiltonian
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{ijk}\,\!}				A complete set of Hermitian matrices in terms of which any Hermitian matrix can be expanded
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i\,\!}				Form a basis for the stabilizer of the system
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i g_i S_i\,\!}				An arbitrary linear combination of those stabilizer elements
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ijk}\,\!}				A set of real numbers
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i\,\!}				Arbitrary coefficients
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_n \,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_n \,\!} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_n \,\!}				The Pauli x-operation, y-operation, and z-operation on the nth qubit (see Eq. 8.27)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{X} \,\!}				A logical Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \,\!} operation (see Eqs. 8.28 - 8.34)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Z} \,\!}				A logical Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z \,\!} operation (see Eqs. 8.29 - 8.35)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Y} \,\!}				A logical Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y \,\!} operation (see Section 8.5.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{ZZ} \,\!}				A logical two-qubit entangling gate (see Eq. 8.30)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{ex}^{i,j} \,\!}				The Heisenberg exchange interaction Hamiltonian between two qubits labelled Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \,\!} and $j\,\!$ (see Eq. 8.31)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{ij} \,\!}				An operator resulting from the exponential of the exchange operation between qubits Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \,\!} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j \,\!} for a time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/4\,\!} (see Section 8.5.2 and Eq. 8.32 - 8.35)
		Chapter 9 - Dynamical Decoupling Controls
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U(t) \,\!}				A unitary matrix (see Eqs. 9.3 - 9.6)
${\mathcal {T}}\,\!$				The time-ordered exponential (see Eqs. 9.5)
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \,\!}				Number of different controls to be used
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_n \,\!}				A given control
$H\,\!$				Free evolution given by Eq. 9.1
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_i \,\!}				The Hamiltonian for the free evolution (see Eq. 9.13)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\,\!}				The bath operator (see Eq. 9.13)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_z\,\!}				Indicates a phase error (see Eq. 9.13)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_X(\pi)\,\!}				A decoupling pulse, denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_1 = U_X(\pi)\,\!} (see Eq. 9.14)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_0\,\!}				The identity, denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_0=I\,\!} (see Eq. 9.14)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_k\,\!}				Some particular element of the group (see Eqs. 9.19 - 9.22)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\,\!}				Some constant (as of yet unknown) (see Eq. 9.19 - 9.22)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\lambda_i\}\,\!}				A complete set of Hermitian matrices (see Eqs. 9.23 - 9.25, Section C.3.8, and Section C.6.1)
$a_{i}\,\!$				Set of coefficients (see Eqs. 9.23 - 9.26)
		Chapter 10 - Fault-Tolerant Quantum Computing
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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mbox{Shor}\rangle_{\text{anc}}\,\!}				Shor proposed ancilla state (see Eq. 10.1)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\text{Steane}\rangle_{\text{anc}}\,\!}				Steane proposed ancilla state (see Eq. 10.2)
		Chapter 11 - Hybrid Methods of Quantum Error Prevention
$\left|0_{d}\right\rangle \,\!$				An encoded DFS/NS zero state (see Eq. 11.1)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| 1_d\right\rangle\,\!}				The corresponding DFS/NS encoded one state (see Eq. 11.1)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \psi \right\rangle_S \,\!}				The system (see Eq. 11.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \phi\right\rangle_G\,\!}				The gauge system (see Eq. 11.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{E}\,\!}				An error (see Eq. 11.2)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{R}\,\!}				An error recovery operation (see Eq. 11.2)